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arXiv · 2407.05039

Free-Boundary Monotonicity for Almost-Minimizers of the Relative Perimeter

Gian Paolo Leonardi·Giacomo Vianello
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Abstract

Let $E \subset Ω$ be a local almost-minimizer of the relative perimeter in the open set $Ω\subset \mathbb{R}^{n}$. We prove a free-boundary monotonicity inequality for $E$ at a point $x\in \partialΩ$, under a geometric property called ``visibility'', that $Ω$ is required to satisfy in a neighborhood of $x$. Incidentally, the visibility property is satisfied by a considerably large class of Lipschitz and possibly non-smooth domains. Then, we prove the existence of the density of the relative perimeter of $E$ at $x$, as well as the fact that any blow-up of $E$ at $x$ is necessarily a perimeter-minimizing cone within the tangent cone to $Ω$ at $x$.

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Gian Paolo Leonardi, Giacomo Vianello. 2025-08-01. Free-Boundary Monotonicity for Almost-Minimizers of the Relative Perimeter. https://doi.org/10.4171/ifb%2F544

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